Subgroups — C₄₁×#33638 #23809839

Order 10496.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#2957	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2231	C₂	1	4	yes	maximal, complemented
4	C₄₁×#2230	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₁×#360	C₂	1	1	yes	maximal, characteristic
10	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#568	—	2	1	no	—
12	C₄₁×#544	—	2	1	no	—
13	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#501	—	2	2	no	—
16	C₄₁×#499	C₂²	1	2	yes	complemented
17	C₄₁×#498	C₂²	1	2	yes	complemented
18	C₄₁×#444	—	2	2	no	—
19	C₄₁×#443	C₂²	1	4	yes	—
20	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#428	—	2	1	no	—
22	C₄₁×#427	C₂²	1	1	yes	characteristic
23	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#421	—	2	1	no	—
25	C₄₁×#405	C₂²	1	1	yes	characteristic
26	C₄₁×#365	C₂²	1	4	yes	complemented
27	C₄₁×#365	C₂²	1	2	yes	complemented
28	C₄₁×#365	C₂²	1	2	yes	complemented
29	C₄₁×#364	C₂²	1	4	yes	complemented
30	C₄₁×#360	C₂²	1	4	yes	—
31	C₄₁×#360	C₂²	1	2	yes	—
32	C₄₁×#360	C₂²	1	2	yes	—
33	C₂×#41×C₄₁	—	2	1	no	—
34	#135×C₄₁	—	2	1	no	—
35	#135×C₄₁	—	2	1	no	—
36	#135×C₄₁	Q₈	1	2	yes	complemented
37	#135×C₄₁	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₄₁	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₄₁	—	2	1	no	—
40	C₂×D₈×C₄₁	Q₈	1	1	yes	characteristic, complemented
41	#131×C₄₁	—	2	4	no	—
42	#131×C₄₁	—	4	2	no	—
43	C₂×M₁₆×C₄₁	—	2	1	no	—
44	C₂×M₁₆×C₄₁	—	4	2	no	—
45	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₁	—	2	2	no	—
48	C₄⋊Q₈×C₄₁	—	4	1	no	—
49	#125×C₄₁	—	2	2	no	—
50	#125×C₄₁	—	4	1	no	—
51	C₂²⋊Q₈×C₄₁	—	4	2	no	—
52	#117×C₄₁	—	2	1	no	—
53	#117×C₄₁	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
55	M₃₂×C₄₁	—	2	1	no	—
56	M₃₂×C₄₁	—	2	1	no	—
57	M₃₂×C₄₁	C₂³	1	2	yes	—
58	M₃₂×C₄₁	C₂³	1	2	yes	—
59	C₂×C₁₆×C₄₁	—	2	1	no	—
60	C₂×C₁₆×C₄₁	—	2	1	no	—
61	#108×C₄₁	—	2	1	no	—
62	#108×C₄₁	C₂³	1	2	yes	—
63	#107×C₄₁	—	2	1	no	—
64	#107×C₄₁	—	2	2	no	—
65	#106×C₄₁	—	2	1	no	—
66	#106×C₄₁	—	2	2	no	—
67	#106×C₄₁	C₂³	1	2	yes	—
68	#106×C₄₁	C₂³	1	1	yes	characteristic
69	#106×C₄₁	C₂³	1	1	yes	characteristic
70	#104×C₄₁	—	2	4	no	—
71	#97×C₄₁	D₄	1	2	yes	—
72	C₄×C₈×C₄₁	C₂³	1	2	yes	—
73	#41×C₄₁	—	2	2	no	—
74	#41×C₄₁	—	2	2	no	—
75	#41×C₄₁	—	4	1	no	—
76	#41×C₄₁	—	4	1	no	—
77	C₂×Q₈×C₄₁	—	2	1	no	—
78	C₂×Q₈×C₄₁	—	4	2	no	—
79	C₂×D₄×C₄₁	—	2	1	no	—
80	C₂×D₄×C₄₁	—	4	1	no	—
81	C₂²×C₄×C₄₁	—	4	1	no	—
82	C₂²×C₄×C₄₁	—	4	1	no	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₄₁	—	2	1	no	—
85	Q₁₆×C₄₁	C₂×Q₈	1	2	yes	—
86	SD₁₆×C₄₁	—	2	2	no	—
87	SD₁₆×C₄₁	—	2	2	no	—
88	D₈×C₄₁	—	2	1	no	—
89	D₈×C₄₁	C₂×Q₈	1	2	yes	—
90	M₁₆×C₄₁	—	2	2	no	—
91	M₁₆×C₄₁	—	4	1	no	—
92	M₁₆×C₄₁	—	4	4	no	—
93	M₁₆×C₄₁	SD₁₆	1	2	yes	—
94	M₁₆×C₄₁	SD₁₆	1	2	yes	—
95	C₂×C₈×C₄₁	—	4	1	no	—
96	C₂×C₈×C₄₁	—	4	4	no	—
97	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
103	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₁	—	2	1	no	—
105	C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₄⋊C₄×C₄₁	—	2	1	no	—
107	C₄⋊C₄×C₄₁	—	2	1	no	—
108	C₄⋊C₄×C₄₁	—	4	2	no	—
109	C₄⋊C₄×C₄₁	—	4	2	no	—
110	C₄⋊C₄×C₄₁	—	4	2	no	—
111	C₄⋊C₄×C₄₁	—	4	1	no	—
112	C₂²⋊C₄×C₄₁	—	2	1	no	—
113	C₂²⋊C₄×C₄₁	—	4	1	no	—
114	C₄²×C₄₁	—	4	1	no	—
115	C₄²×C₄₁	C₂×D₄	1	2	yes	—
116	C₁₆×C₄₁	—	2	2	no	—
117	C₁₆×C₄₁	—	2	2	no	—
118	C₂³×C₄₁	—	4	1	no	—
119	C₂³×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₄₁	—	2	2	no	—
121	Q₈×C₄₁	—	4	1	no	—
122	Q₈×C₄₁	—	16	2	no	—
123	D₄×C₄₁	—	2	2	no	—
124	D₄×C₄₁	—	4	2	no	—
125	D₄×C₄₁	—	4	2	no	—
126	D₄×C₄₁	—	4	1	no	—
127	C₂×C₄×C₄₁	—	2	2	no	—
128	C₂×C₄×C₄₁	—	4	1	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	1	no	—
132	C₂×C₄×C₄₁	—	4	1	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	16	1	no	—
135	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₄₁	—	4	2	no	—
142	C₈×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₄₁	C₂×SD₁₆	1	4	yes	—
145	C₈×C₄₁	C₂²×Q₈	1	1	yes	characteristic
146	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
147	#33638	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₁	—	4	1	no	—
149	C₂²×C₄₁	—	4	2	no	—
150	C₂²×C₄₁	—	16	1	no	—
151	C₂²×C₄₁	#476	1	1	yes	characteristic
152	C₂²×C₄₁	#476	1	1	yes	characteristic
153	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₄×C₄₁	—	4	2	no	—
155	C₄×C₄₁	—	4	2	no	—
156	C₄×C₄₁	—	8	1	no	—
157	C₄×C₄₁	—	8	2	no	—
158	C₄×C₄₁	—	8	1	no	—
159	C₄×C₄₁	#476	1	1	yes	characteristic
160	C₄×C₄₁	#476	1	1	yes	characteristic
161	C₄×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₄×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
163	#3163	C₂×C₄₁	1	1	yes	characteristic, complemented
164	#2957	C₂×C₄₁	1	1	yes	characteristic
165	#2231	C₂×C₄₁	1	4	yes	complemented
166	#2230	C₂×C₄₁	1	4	yes	complemented
167	#2155	C₂×C₄₁	1	1	yes	characteristic, complemented
168	C₂×#365	C₂×C₄₁	1	1	yes	characteristic, complemented
169	#2153	C₂×C₄₁	1	1	yes	characteristic, complemented
170	#2145	C₂×C₄₁	1	1	yes	characteristic
171	C₂×#360	C₂×C₄₁	1	1	yes	characteristic
172	C₂×C₄₁	—	2	1	no	—
173	C₂×C₄₁	—	2	1	no	—
174	C₂×C₄₁	—	2	1	no	—
175	C₂×C₄₁	—	8	2	no	—
176	C₂×C₄₁	C₂×#476	1	1	yes	characteristic
177	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
178	#568	—	2	1	no	—
179	#544	—	2	1	no	—
180	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
181	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
182	#501	—	2	2	no	—
183	#499	C₂²×C₄₁	1	2	yes	complemented
184	#498	C₂²×C₄₁	1	2	yes	complemented
185	#444	—	2	2	no	—
186	#443	C₂²×C₄₁	1	4	yes	—
187	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
188	#428	—	2	1	no	—
189	#427	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
191	#421	—	2	1	no	—
192	#405	C₂²×C₄₁	1	1	yes	characteristic
193	#365	C₂²×C₄₁	1	4	yes	complemented
194	#365	C₂²×C₄₁	1	2	yes	complemented
195	#365	C₂²×C₄₁	1	2	yes	complemented
196	#364	C₂²×C₄₁	1	4	yes	complemented
197	#360	C₂²×C₄₁	1	4	yes	—
198	#360	C₂²×C₄₁	1	2	yes	—
199	#360	C₂²×C₄₁	1	2	yes	—
200	C₄₁	#33638	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#41	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	Q₈×C₄₁	1	2	yes	complemented
205	#135	C₂³×C₄₁	1	2	yes	—
206	C₂×Q₁₆	Q₈×C₄₁	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	Q₈×C₄₁	1	1	yes	characteristic, complemented
209	#131	—	2	4	no	—
210	#131	—	4	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	4	2	no	—
213	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	D₄×C₄₁	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	C₂³×C₄₁	1	2	yes	—
226	M₃₂	C₂³×C₄₁	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂³×C₄₁	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#106	—	2	1	no	—
234	#106	—	2	2	no	—
235	#106	C₂³×C₄₁	1	2	yes	—
236	#106	C₂³×C₄₁	1	1	yes	characteristic
237	#106	C₂³×C₄₁	1	1	yes	characteristic
238	#104	—	2	4	no	—
239	#97	D₄×C₄₁	1	2	yes	—
240	C₄×C₈	C₂³×C₄₁	1	2	yes	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Q₈×C₄₁	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Q₈×C₄₁	1	2	yes	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	SD₁₆×C₄₁	1	2	yes	—
262	M₁₆	SD₁₆×C₄₁	1	2	yes	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
267	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
268	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
269	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
270	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
271	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×D₄×C₄₁	1	2	yes	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	16	1	no	—
303	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
304	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
305	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
306	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
307	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
311	C₈	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
312	C₈	C₂×SD₁₆×C₄₁	1	4	yes	—
313	C₈	C₂²×Q₈×C₄₁	1	1	yes	characteristic
314	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	C₄₁×#476	1	1	yes	characteristic
319	C₂²	C₄₁×#476	1	1	yes	characteristic
320	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	C₄₁×#476	1	1	yes	characteristic
327	C₄	C₄₁×#476	1	1	yes	characteristic
328	C₄	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
329	C₄	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₄₁×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₂×#135×C₄₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ C₄₁×#405 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄₁×#3163	C₂	semidirect
3. C₄₁×#2231	C₂	semidirect
4. C₄₁×#2230	C₂	semidirect
5. C₄₁×#2155	C₂	semidirect
6. C₂×C₄₁×#365	C₂	semidirect
7. C₄₁×#2153	C₂	semidirect
16. C₄₁×#499	C₂²	semidirect
17. C₄₁×#498	C₂²	semidirect
26. C₄₁×#365	C₂²	semidirect
27. C₄₁×#365	C₂²	semidirect
28. C₄₁×#365	C₂²	semidirect
29. C₄₁×#364	C₂²	semidirect
36. #135×C₄₁	Q₈	semidirect
38. C₂×Q₁₆×C₄₁	Q₈	semidirect
40. C₂×D₈×C₄₁	Q₈	semidirect
147. #33638	C₄₁	direct (N × H)
163. #3163	C₂×C₄₁	semidirect
165. #2231	C₂×C₄₁	semidirect
166. #2230	C₂×C₄₁	semidirect
167. #2155	C₂×C₄₁	semidirect
168. C₂×#365	C₂×C₄₁	semidirect
169. #2153	C₂×C₄₁	semidirect
183. #499	C₂²×C₄₁	semidirect
184. #498	C₂²×C₄₁	semidirect
193. #365	C₂²×C₄₁	semidirect
194. #365	C₂²×C₄₁	semidirect
195. #365	C₂²×C₄₁	semidirect
196. #364	C₂²×C₄₁	semidirect
200. C₄₁	#33638	direct (N × H)
204. #135	Q₈×C₄₁	semidirect
206. C₂×Q₁₆	Q₈×C₄₁	semidirect
208. C₂×D₈	Q₈×C₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₁×#3163	2
C₄₁×#2957	2
C₄₁×#2231	2
C₄₁×#2230	2
C₄₁×#2155	2
C₂×C₄₁×#365	2
C₄₁×#2153	2
C₄₁×#2145	2
C₂×C₄₁×#360	2
#33638	41

Subgroups — C41×#33638 #23809839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33638 #23809839